Structured Low-Rank Matrix Factorization: Global Optimality, Algorithms, and Applications

TL;DR

This paper introduces a structured low-rank matrix factorization method that is scalable for large datasets, captures complex data structures, and guarantees global optimality under certain conditions, with practical algorithms and applications.

Contribution

It proposes a non-convex matrix factorization approach with regularization that ensures global optimality for large factors, addressing scalability and structural complexity.

Findings

Effective in neural calcium imaging video segmentation

Improves hyperspectral compressed recovery results

Provides bounds on solution optimality

Abstract

Recently, convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it challenging to apply them to large scale datasets. Moreover, in many applications the data can display structures beyond simply being low-rank, e.g., images and videos present complex spatio-temporal structures that are largely ignored by standard low-rank methods. In this paper we study a matrix factorization technique that is suitable for large datasets and captures additional structure in the factors by using a particular form of regularization that includes well-known regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is non-convex, we show that if the size of the factors is large enough, under certain conditions, any local minimizer for the factors yields a global minimizer. A few practical algorithms are also provided to solve the matrix factorization problem, and bounds on the distance from a given approximate solution of the optimization problem to the global optimum are derived. Examples in neural calcium imaging video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets.